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We show that if a graph $G$ satisfies certain conditions then theconnectivity of neighbourhood complex $\mathcal{N}(G)$ is strictly less thanthe vertex connectivity of $G$. As an application, we give a relation betweenthe connectivity of the neighbourhood complex and the vertex connectivity forstiff chordal graphs, and for weakly triangulated graphs satisfying certainproperties. Further, we prove that for a graph $G$ if there exists a vertex $v$satisfying the property that for any $k$-subset $S$ of neighbours of $v$, thereexists a vertex $v_S \neq v$ such that $S$ is subset of neighbours of $v_S$,then $\mathcal{N}(G-\{v\})$ is $(k-1)$-connected implies that $\mathcal{N}(G)$is $(k-1)$-connected. As a consequence of this, we show that:(i) neighbourhoodcomplexes of queen and king graphs are simply connected and (ii) if $G$ is a$(n+1)$-connected chordal graph which is not folded onto a clique of size$n+2$, then $\mathcal{N}(G)$ is $n$-connected.

Vertex cut of a graph and connectivity of its neighbourhood complex